While I agree that the Western states have been controlled by anti-Christians in a subtle way (at times there is not much subtlety, the point is that the typical Christian can be raised not realizing that the system of government in their country is fundamentally anti-Christian), the fact is that the Communists were bent on the total destruction and subversion of Christianity in a very militant way. Communists with ties to Russia were not just working against the Church in Russia, but everywhere.
By dominating Eastern Europe, the Communists were in a position to blackmail priests and bishops like never before.
The subversion of the priesthood in the Western world was a goal of the Communists.
When I see the phrase "Russia will spread her errors" I see it as meaning Russia will spread the Revolution. The Revolution will try to militantly persecute or craftily subvert the Church everywhere.
To me the question comes down to what happened in 1958 and 1963.
Is it possible, some sort of horrific threat was made that caused those who should have defended the Faith to lose their nerve?